### Abstract

We consider a blocking problem: fire propagates on a half plane with unit speed in all directions. To block it, a barrier can be constructed in real time, at speed σ. We prove that the fire can be entirely blocked by the wall, in finite time, if and only if σ > 1. The proof relies on a geometric lemma of independent interest. Namely, let K ⊂ R
^{2}
be a compact, simply connected set with smooth boundary. We define d
_{K}
(x, y) as the minimum length among all paths connecting x with y and remaining inside K. Then d
_{K}
attains its maximum at a pair of points (over(x, ̄), over(y, ̄)) both on the boundary of K.

Original language | English (US) |
---|---|

Pages (from-to) | 133-144 |

Number of pages | 12 |

Journal | Journal of Mathematical Analysis and Applications |

Volume | 356 |

Issue number | 1 |

DOIs | |

State | Published - Aug 1 2009 |

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### All Science Journal Classification (ASJC) codes

- Analysis
- Applied Mathematics

### Cite this

*Journal of Mathematical Analysis and Applications*,

*356*(1), 133-144. https://doi.org/10.1016/j.jmaa.2009.02.039

}

*Journal of Mathematical Analysis and Applications*, vol. 356, no. 1, pp. 133-144. https://doi.org/10.1016/j.jmaa.2009.02.039

**The minimum speed for a blocking problem on the half plane.** / Bressan, Alberto; Wang, Tao.

Research output: Contribution to journal › Article

TY - JOUR

T1 - The minimum speed for a blocking problem on the half plane

AU - Bressan, Alberto

AU - Wang, Tao

PY - 2009/8/1

Y1 - 2009/8/1

N2 - We consider a blocking problem: fire propagates on a half plane with unit speed in all directions. To block it, a barrier can be constructed in real time, at speed σ. We prove that the fire can be entirely blocked by the wall, in finite time, if and only if σ > 1. The proof relies on a geometric lemma of independent interest. Namely, let K ⊂ R 2 be a compact, simply connected set with smooth boundary. We define d K (x, y) as the minimum length among all paths connecting x with y and remaining inside K. Then d K attains its maximum at a pair of points (over(x, ̄), over(y, ̄)) both on the boundary of K.

AB - We consider a blocking problem: fire propagates on a half plane with unit speed in all directions. To block it, a barrier can be constructed in real time, at speed σ. We prove that the fire can be entirely blocked by the wall, in finite time, if and only if σ > 1. The proof relies on a geometric lemma of independent interest. Namely, let K ⊂ R 2 be a compact, simply connected set with smooth boundary. We define d K (x, y) as the minimum length among all paths connecting x with y and remaining inside K. Then d K attains its maximum at a pair of points (over(x, ̄), over(y, ̄)) both on the boundary of K.

UR - http://www.scopus.com/inward/record.url?scp=63349108339&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=63349108339&partnerID=8YFLogxK

U2 - 10.1016/j.jmaa.2009.02.039

DO - 10.1016/j.jmaa.2009.02.039

M3 - Article

VL - 356

SP - 133

EP - 144

JO - Journal of Mathematical Analysis and Applications

JF - Journal of Mathematical Analysis and Applications

SN - 0022-247X

IS - 1

ER -